Best way to handle Serial data from a PC?

I'm not sure - I guess it depends how much flexibility you want on the PC. For instance if you want to be able to draw bitmaps you might as well just draw one big bitmap which is the screen.

You don't have to use JSON though - you could just send the data as binary. Only issue then is 'framing' so you know when the start of the data is, but you could do that by just having a timeout.